Download - EΠΑΝΑΛΗΨΗ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ.pdf
-
E
-
1
z : ( ) ( )4Re 2Re , 1z zz
+ =
. z . . ( )Re 0z , :
i. 4w zz
= + 4 4w .
ii. 3 4c z i= + + .. , c .
. 1 2,z z 3z (1) ,
1 2 2 3 3 1 1 2 32z z z z z z z z z+ + = + +
2
3( ) 4( ),zz z z i z z z C+ = + (1) : . z
. z 1z
. , R , 1z 21 0
4z z + + =
. 0z (1), 2012
20120 4 3 5 , 55
z i w iw i
+=
,
w (0,5) 2 1 = .
30
+ 10
-
3
z C , 1z = 1z a+ = , a R . : . 0 2a
. 2 2( )2
aRe z =
. 2 21 3z z a + =
. 2133 14
a z z a +
4
,z w 2 2| | 1,| | 3z zw w zw+ = + = . . | | 2z w+ = . . z w , . . z w . . ,z w .
5
2 2(cos ) (5 4sin ) 0, [0, ]z t z t t + = . : . 1 2,z z
. 1 2| |z z
. 1 2| |z z+
6
( 1) , [0,1]z t t i t= + . . z . | |z
2 2( 2) ( 1) ,w k k i k= + + , : . w . | |w . | |z w . | |w | |z w [0, 4]k
-
7
. 2 1 0w w+ + = . 1 2,z z
2 21 1 2 2 0z z z z+ + =
i. : 1 2z z=
ii. : 1 2 1 2z z z z+ = =
iii. *N 1 2 0z z + , 1 2
1 2
z zuz z
=
+ .
8
1 2 3, ,z z z , ,A B ,
: 1 2 32 3z z z+ = 1 3 21, 2z z z= = =
. 1 2Re( ) 0z z =
. i. 2 2 2
1 2 1 2z z z z = +
ii. OAB .. 2 3Re( )z z 1 3Re( )z z. i. , ,A B .
ii. A B
9
1 2,z a bi z c di= + = + , , ,a b c d 1 2 1z z= = .
2 1 22 2 0x z z x + = 1 2,x x . :
. 1 2,x x .
. 1 2 2x x= =
. 2 2
1 2 1 24 8x x z z + =
. 1 22 1
x xwx x
= + .
-
10
( ) ( )6 8 , 0z t t i t = + + + .. 6 8z i . . z . . z . 0t , z 1 3
. 0t = R , 11
w zz
= + ++
.
11
2( )(1 )x y iz
x yi+ +
=+
*x, y R .
. ( )2 2
2 2
22 x y xyRe zx y+ +
= +
( )2 2
2 22x yIm zx y
=+
*x, y .
. z . . z . . z .
12
z 2| 1| | 9 20 |z z z = + | 4 | , 0z = > . . :
i. 2 2| | 16
4zz z + + =
ii. 2 2 2 2(1 ) | | (5 1)( ) 25 1z z z + + = iii. | | 2z
. z . . z .
13
,z w C 2 22 2(3 4) 5 0, (4 3 ) 5 0z i iz w i iw+ + = + + = .
. 2 2(4 3 ) 5 ( ) 0z w i i z w+ + + + = .
. q 1 1qz
= + .
-
14
,z w C 1zw = .
, , ,a b c d R 2 2 2 2 0a b c d+ + > ( ) ( ) ( 1) ( 1) 0a z z ib z z c zz d zz+ + + + + = , : . z . z , w . . z , w
.
15
, *a C 1 2,z z 2 2 0z az + + = . :
. | | | | 1a = = 1| | 2z 2| | 2z .
. 1 2| | | |z z= , a
.
. 2a R , 1
2
zz
, 1 2| | | |z z=
16
*22( 2 ),z z Cz
= (1)
. 1z 2z (1)
. v , 1 2 0v vz z+ =
. x y ,
2011 162 21 2
1 1 1( )i ix yi z z
+ = + ++
. z , 2 41 2z z z z =
. 0z .
. 7z i+ .
-
17
( ) ( )z k t k t i = + t R 1k > . . z .
. w ( 1)y x k= , k 5 2| | 12min
z w =
. k () z | |z z . k () | 3 4 |w i min + . u ( 1 ) ( 1 )u m t m t i = + + + , m u . . ,k m () () , | |z u .
18
z, w 2 24 | | 2 1,| | 2 3z zw w zw = = . . | 2 | 2z w = . . z w , . . | 6 |z w+ . . z w .
19
z ( )| 2 | | 1|
if zz z
=
. z ( )f z . . | ( ) | 1f z . ( )f z i= , :
i. N | 1| ( ) 1z Re z + =ii. ( )Re z .iii. z , z
.i.
20
. ( 3 )( 4 2 )z i z i+ + +. 2 (1 3 ) 14 2 0z i z i + = (1). 1 2,z z (1) 1Re( ) 0z > , ,A B 1 2,z z 3 3z i= + . AB . M( )z z
2 2 2( ) 2( ) 2( ) 30MA MB M+ = +
-
21
1 2,z z 12
z zz
= 1 2, ( ) 0z Im zz
+ = .
. z . . *N 2 21 2z z
= .
. 95 94 ... 1 0z z z+ + + + = . 2z 2 1y x= + , 1z , . . OAB , , ,O A B 1 20, ,z z . . OAB () .
22
{2 }z C i 3 8( )
2z if zz i+
=
. 2z i , 2( ) 2 4f z z iz= +
. (1 )f i+
. 2( ) 2 4f z z iz= +
. ( ) 2 6f z iz z= +
. 1z = , ( )f z 7 =
23
z ( ) ( )2005 2008 1 1z z = .. z
. 2z z= . (1)
. z ( )Im 0z > , (1) zwz
+
=
R .
i. w
ii. , w
-
24
1 2,z z 1( ) 0Im z > 1 2 2 1| | | | 40(1)z z z z+ = 1 2 25(2)z z = .
. 1z 2z .
. 1w z i= , .
. *N 1 2z z z = . 1 2,z z z
.
25
2 0, ,z z R + + = 12zi
= 2z
. , R 2z
. v R , 1 2 16v vz z i =
. z
2 2
1 2 16z z z z + = (1)
. z (1) , 4 4z i
26
, 0z w 0zw zw+ =
. 2010z
w
.
. ,z w .
. z w z w = + .
. 2z w iw z+ =
i. zw
( )1,0 .
ii. 2012z
w
.
27
z : 2 2 2 21 2 | | | 1| 2 | 1|z z z+ = + + +. z . . 2 2( 1) ( 1) 0z z zz+ + + =
-
. 2 1 0z z+ + = . z . . 2012 2014 2013A z z= + +
28
z 2z i 2 4( )
2zf zz i+
=
(1)
. Im( (1 ))f i+ . z , ( )f z R
. ( ) 2f z z i= + . z , ( 5 ) ( ) 10f z i f z i + + = (2) . z (2) ,
. 1z 2z (2) , 1 28 10z z
29
1 2z , z 2z z+9=0 , R 1 2z , z R .
. .
. ( )17 171 2z z R+ . . 1 2z , z .
. 1 22 1
z z 2z z
+ = .
. =0 ( )1Im 0z > z 1 2z z 4 z z = + .
30
. , , : 3 23 4 2 0z z z + =
. , : 3 24 3 2z z z+ = + 10 32 32 0z z + =
-
31
f , [ ],a b , 0a > ( ),a b . : ( ) ( )1 2z a if a z b if b= + = +. 1 2 1 2z z z z+ = , ( )1 ,x a b , ( )1 0f x =. A B , A B , : 1 2 1 2 100Az z Bz z+ = . :
i. 1 2z z
ii. ( ) ( )f a f ba b
=
iii. ( ),ox a b , ( )( )o
oo
f xf x
x =
iv. fC
32
f : [ , ] R ( ) 0f > > , ( )( )
ifzif
+
=
. :
. ( )f x x= ( , ) . 0 ( , )x 0( ) 1f x z .
. i. f z , ii. .
.
Bolzano f [ 3, 3]
. ( ) 23 ( 2 1)3
w f z i= + + , w
-
34
z x yi= + , 22012 ( ) 1 2012 2012 3 0
21Im z i
z+ + =
:
. ( , )M x y z . . z ; . z , ( ) 0Im z < .
35
f R , ( ) 0f x x R 1
( ) 0z
f x dx = f(1) = 1 .. ( ) 0f x > . z .
. ( )( )
3
2
3lim
3xz z x xz z x x+ +
+
. f 'x x 0x = 1x = 2z z+ ,
20
( ) 3 6 6x
f t dt x x= + (0,1)
36
[ ], f ( )2z if = + ( )2w if = 0 ( ) ( ) 0f f . w z w z+ <
( ) ( ) ( )f f f < < . . ( )0 ,x ( )0 0f x =. ( )1 ,x ( ) ( )1f x f =
37
. 1 2,z z 1 2 1 2z z z z+ .
1 2w z z= .
. ( )1 1f xz i= + ( )( )2 1z f x i= + +
() f R ( )0 0f = ( )0 0f . 2 3< <
-
. g ( ) 1 2( )g x Im z z= Rolle [ ], ( )
( )( )( )
11
f
f
fefe
+=
+, ( ) 1f
38
f : R R (0, 2)A .
( ) ( )z f x f x i= + 2( ) ( )w f x f x i= 2( 1)xz e= +
. f . . z x R . . ( ) Re( )g x z w= .
39
:f R R 12
z C
2 2( ) 2 ( )f x x xf x+ =
x 0
( ) | 2 |,|
m2 1|
lix
f x zl lx z
= =
.
. : i. | 2 | | 2 1|z z = ii. z .
. 20( )lim
x
f xx x
.
. ( ) ( )g x f x x= ( ,0) (0, )+ . f .
. 3(| 3 4 | 5) 10z i x x+ + = + [ ]1,2
40
( ) ( )1 3 3 3z x x i = + + + , [ )0,2x .. M z ( )C .
. x z .
z .
-
. 1 2,x x x z
1 2,M M z . :f R R
1 2,M M . f
'x x ( )C .
http://www.mathematica.gr/forum/viewtopic.php?f=51&t=21713http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=4316http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=129http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775 -
1
. f [0,1] 0 (0,1)x 00 0
1 1( )1
f xx x
=
.
. f R 1 ( ) xx f x e+ x R . :
i. f 0 0x =
ii. f R , 0 ( 1,1)x , 0 0( )
2004f x x= .
iii. lim ( )x
f x+
2
f : 55 3( ) 2 2f x x z x z= + x R *z C .
. f . . f .
. ( ) 0f x = (0, )z
. 5
30
( ) 2lim 1x
f x zx
+= , z .
3
R f :g R R 2( ) ( ) ( ) 1xg x f x e g x= + x R | (0) | 1f < .
. (lim )x
f x
. (lim )x
g x
30
http://www.study4exams.gr/ -
. (0) 1g
. ( 1)( )[ ( ) ] [ ( 1) ]x xx e x g x e x f x x + + = + [0,1) .
4
:f R R ( ) 112
f = ( ) ( ) ( )3 3f xy f x f f y fy x
= +
*,x y R . :
i. ( ) 132
f =
ii. ( )3 ,f f x x Rx
=
iii. ( ) ( ) ( )2f xy f x f y= ( )2 14
f x = x R
. f .
5
f : R R g : R R x R ( ( )) 2 ( )f f x g x x= . g R . . ( ) ( ) ( )h x f x g x= . 0x R 0 0( )f x x=
i. fC gC .
ii. 0 0 0( ( 2)) 2 ( 2) 2f f x x x x f x x+ + + = + +
iii. 0 0( (ln 1)) ln 1f f x x x x+ + + +
( ) 0f x = . . (1)f . f .
-
. 2( 2) ( ) (5 6)f x f x f x + = . ( ) 0f x < 1x > , f (0, )+ .
7
( ) 2 ln( 2 1)f x x= + . . . f .
. f 1f
. 1( ) 2f x = . fC y x=
8
:f R R : ( ( )) (2 )f x f x y f x y+ + = + ,x y R . : . (0) 0f = . ( )( )f f x x = x R . f 1 1 . f R . . ( )f x x= x R
9
( ) (1 ) (1 )x xf x ln e ln e= + . fD
. ( )f x. f
. 1f . 0m < ( )f m m= . ( ) ( ) , 0g x f x x x= < , ( )g x . ( ) ( 1) 1f x f x < + . ( ) ( )h x ln x= , c R ( ) ( )f c h c=
. : 3 2
2
( 1) 6lim( 3) 2x
f x xAf x x + +
=
, ( )1( ) ( )lim f x f xxB e e
=
-
10
f ( ) lnf x x x= + . . f . . ( ) 1f x = . f ( ) 0f x = . . f f .
. 1( )f x x =
. 1( ) 1f x x >
11
f 2 ( ) ( )f x f x x =
. 2 ( ) ( )f x x f x
. ( )f x x
. 0
( )lim ( )xf x
f x
. 0
( )lim x
f xx
12
23( ) 3 30 95 (3 5), ,
4 4g x x x x x R R = + + + 4 5( ) ,
3xh x x R=
. f goh= x R 2( ) ( )( ) 5 10f x goh x x x x= = + +
. R lim ( )x
f x+
. R lim ( )x
f x+
42
( ) 2lim14x
f x xx x
++
. R 3
lim( )x
xf x x
+ +
-
13
:f R R ( )( ) ( ) 1f x f y f x y+ = + + ,x y R . : . f .
. ( ) ( ) 12 ( ) 1f x f x f x= + +. f . . ( 1) 0f =
14
, < :f R R ,
: ( ) 2f = , ( ) 2f = ( ) 2012f x < x R .. ( ) ( )2x f x f = + ( ]0, + . f [ ], = :
i. ( , ) ( )f = + . ii. fC f 2y x= ( ),ox .
. 2( ) 4
1xxf x xlim
x
+ +. :h R R ( ) ( ) 2004f x h x x = , x R .
( ) 0f x = 1 2, . ( ) 0h x = ( )1 2,
15
1 1 , :f g R R 1( ) ( )( ) 8f x fog x = 13( )( ) 2( )( ) 10 7fog x fog x x+ = x R
. ( ), ( )f x g x
. ( ) :h x R R 2( ( ( ))) 4 2xh g f x e x= x R i. ( )h xii. ( )h x .
iii. 22
3 2 6xxe e x x
ee >
iv. : ( )lim
x
h xx+
, 2( 1)lim
1xxh xe+
++
, 1 3 2
2
( ) 2 3lim5x
h e x xx x
+
+ ++
. N ( )h x lnx= .
-
16
( ) ( ): 0, 0,f + + ( ) ( )2 1 0f x lnf x lnx+ = .. (1)f
. ( ) ( )1f x fx
=
17
:f R R ( ) 1f x ( ( ) )( ( ) 3 )f x k f y k k + = ,x y R k R . . k . . k f .
18
( ) : (0, )f x R + ( ) ( ) ( )f x f y f x y= + 0ox = (1)f e= .
. 1(0) 1, ( 1)f f e= =. N R . . :
i. ;ii. lim ( )
xf x
lim ( )
xf x
+
. (0,1)ox 1 1 13 ( ) (2 ) (3 ) (4 )of x f f f = + +
. : 10
lim ( )x
f x+
, 1lim ( )
xf x
+ ,
0
( 1 )lim( )x
f xf x +
. 1 1 1( ) ( ) ( )f ab f a f b = + , 0a b >
. 1 2
1
( )lim( )x
f xf x
+
. 1 10 : ( )o o ox f x x > =
. 1
1
( )( )lim( )x
f f xf x
-
19
( ) 3 7 5f x x x= + .. :
i. f 1 1
ii. ( ) 0f x = ( )0,1
. ( )( )
2
3, 1
13 , 1
f xxg x x
a a x
=
+ =
, *a R , ( )g x 1ox =
. i. ( )limx
f x+
( )limx
f x
ii. ox R , ( ) 7of x =iii. ( ), 1k k + , ox R , k
. :
i. ( )
4limxf x x
x
+
ii. , ( ) ( )3 5 2 6f f =
20
f [ ]0,1 2 2(0) (1) 13 6 (0) 4 (1)f f f f+ + = + .
. N : i. f .
ii. 1x 2x (0,1) :
. f 3y x= 1x .
. 212 ( ) 3 (1/ ) 4 (1/ ) 5 (1/ 2)f x f e f f= + +
. 1( ( 4) 1) 3f f lnx + > . ( ) ( )z f x if x= + [0,1]x
i. z .ii. | 5 |z .
21
g 0y > 2( ) lim 1x yyxg x
e x+=
+ +
-
22
f [1,4] : ( ) 0f x [1, 4]x (1) 0f > (1) (2) (3) (4)f f f f= : . ( ) 0f x > [1, 4]x ,
. 2( ) ( ) (1) (2)g x f x f f= (1, 2) . . f .
23
. ( ) ( )1
lim limo
ox x hf x l f x h l
= =
. ( ): 0,f R+ ( ) ( ) ( )f xy f x f y= + , ,x y R . :
i. f 1ox = , f ( )0,+ii. f ox a= , ( ) ( )0,1 1,a + , f ( )0,+
iii. f 1ox = ( )
1lim 1
1xf xx
=
, ( ) ( ) 1lim
x a
f x f ax a a
=
,
( ) ( )0,1 1,a +
24
:f R R : ( ) ( ) ( )f x y f x f y + = + , ,x y R ,R ().
. ( )0f =. ( ) ( ) ( )f x y f x f y = + ,x y R .. ( ) ( ) ( )1f x f x = x R , *N .. ( )f x = R , f 1 1
( ) ( ) ( )1 1 1f x y f x f y + = + ( ),x y f R. 0x > ( )f x > , f R
( ) ( )1 2 12 ( ) (3 1)f f x f f x < + .. f 0ox = lim ( )x f x
-
25
f : R R 3 ( ) 5 ( ) 0f x f x x+ + = x R . f . f
. f R 1f . f R. f R . ( 19) 1f x x = +
. 1
0
( )limx
f xx
26
: Rf R , x R : 2 2( ) 1x f x x< < + .
. fC 2y x= ( )0 0,1x . . f :
i. 1 1( ) 1( ) x
g xf x e
= + , x R
ii. ( ) ( )x xe f x e f x+ = ( )0,2 .
. 20
1lim lnx
x f xx+
+ .
27
: (0,1)f R 0
( ) 5lim 3x
f xx
=
22 ( 1) ( 1) ( ) 1x x f x x (0,1)x . . ( ) ( ) ln 3,g x f x x= (0,1)x .
. ( ) 3 ( ) f xh x e =
, ( )0,1ox . ( )3 f xxe e+ = ( )0,1 , R
-
28
:f R R ( )f R R= R,x y 1( ) ( )f x f y x y
,
( )0,1 . :. f ,
. 1 1( ) ( )f x f y x y ,x y R ,
. 1( )f x R ,
. 1( )f x x = R .
29
, :f g R R ( ) ( ) ( ) ( )2 2 1 2 2f x g x xf x g x+ + = + , x R .
. ( )( ) ( )( )2 2 21f x x g x x + = , x R .. f g 0ox =
. ( )
2limxf xx+
. g R , ( ) 2g x x= R
30
,f g ( ) ( )2f x ln x x= ( ) ( )1g x ln x= .. h ,f g
. 1h . ( ) ( ) ( )( ) ( )1 11 2h x h x ln h ln eh e + = +
- http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775
-
1
:f R R , ( ) 0f x x R ,
( ) ( )1
2
0
1 tf x x dtf xt
= + , x R . ( ) 0f x > x R. f . f x
. ( ) ( )x
x
tg x dtf t
= x R .
. N ( ) ( )f b f a b a ,a b R
2
11
2
( )x
tf x e dt= .. . . f . . fC (2, (2))A f
. ( ) 2f x ex e 1x > . E fC , 'x x 2x =
4x = , 2E e
60 A
-
3
f :[0,1] R
1
( ) ( ) ( )x
f x xf x f t dt = + [0,1]x
. f [0,1]
. 0 (0,1)x 1
00
( ) (1) ( )f x f f t dt = +
. ( ) ( )x
f x x f t dt c= + c R (0) (1)f f=
2
(1)( )1
ff
=+
1
. f (0,1) ,
4
2 201( ) ln( 1)
1x
f x x x dtt
= + + +
. f .
. 34
20
10 ln 21
dtt
< 0,2
20
:f R R ( )( )
0
1 1f x
te dt x+ = x R .
. f . f . f x R . fC , 'x x
1x = 1x e= +
. ( ) ( ) ( ) 11 xx f x f x < < 1x >
-
21
( ) 3xf x e x= ln
1
( 3 )( )3
x t t
te e tg x dt
e t+
=
. f. 1 2,x x ( ) 0f x =
0,3e
( )ln 3,2
. g. g. g , ga b D a b
. ( )1
0
1f x dx =
. ( ) ( )2
0
1x
f t dt xf x= .
24
f : (0, ) R+ ( ) ( ) 1x
e
f t dt f xx
= 0x > .
: . f. ( ) fC
. fC 'x x ( )
.
2
1
( )lim( )
x
xx
f x dtf t+ .
25
f ,g R
01
( )
( )( ) 0
x
x
f t dt
f t dtg t dt >
{0,1}x R g(x) g(2 x) 2+ = g(x) 0 x R .
1
0( ) ( )
x
xf t dt f t dt> .
. 1
0( ) 0f t dt = .
. ( )f 1 0= ( )f 0 0= .
. 1
( ) ( ) ( ) ( )x
f x f t dt f x f x= ( )0,1 f .
. 1
( ) ( )x
f t dt xf x= ( )0,1. 2 ( ) ( )f x xf x= ( )0,1 f .. ( )g x x x
0x = 2x = .
-
26
f R 2
2
1
( ) 1x f tx dt x
x x { }0x R
. 3
( ) ( )x
xx f t dt = .
. ( )f 1 1= ( )f -1 1= . ( )x 0x = (0) 0f = . f ,
.
27
:f R R 32
3 ( ) 6 63
x
x
x tf x f dt x x
= + x R .
. f . . f . . f 1 1 . .
f, 1fC , 'x x 22 , 2x xe
= = .
28
f (0, )+ 2 11 1 1( ) 22 2
x xf x f dtx t t
= + 0x > .
. f (0, )+ ( )f x ( )f x .
. 2( ) ln ( )g x x x f x= + 0x > (0, )+ . . f (0, )x + . . fC .
. ( )E k fC , 'x x
1,x x k= = (0,1)k . . N
0lim ( )k
E k+
.
-
29
f : (0, +) R x > 0 31
1 2 ( )( )x f tf x dt
t t =
. 2ln( ) , 0xf x xx
= >
. f
. ( )E fC , 'x x 1xe
=
x = 0 > , 0
lim ( )E
+
lim ( )E
+
. , R 3 2
2
lnlim 0x
x x xx
+
+ =
30
g : R R f : R R .
, x R ( ) 0f x > ( ) 0g x > . ( )
0
( )( )
xg x tF x f dtg x
=
.
. x R 0
( ) ( ) ( )x
F x g x f u du= . F , ( ) xf x e= ( ) xg x e= . x R ( )F x x , (0) (0) 1g f = . (1) (2) (2) (1)F g F g<
31
: 0, 2f R 2( )f x x = ( )0, 2x
:[0, 2 ]g ( )( )( ) , 0, 2f xg x xx = [0, 2 ) . ( ) ( ) 0f f = = . (0)g
. : 2
( )2
xf xx
( )0, 2x
. : 0
1( )6
xf x dx
=
-
32
:f R R ( ) 0f x > x R
(0) 1f = .
2 ( ) , 0( )
2 , 0
x
x
f t dt xg x t
ln x
= =
. N ( )( ) (2 )2
g xf x f xln
0x > ( )(2 ) ( )2
g xf x f xln
0x 0 > ( )0
( )x f tf dtx
= . g .
. ( )0,1 1
( ) 2 ( )f t dt f
=
39
( ) ( )70
x
f x t t dt = 0, 2x
. 02
f =
. f , ,
. 1n > , ( )11 ffn n
-
41
:[ , ]f R ( ) 2 ( )f f = : 2( ) 2 ( ) 4 ( ) 4f x f x f x = + [ , ]x .
: . f . . ( ) 0f > .
. ( ) 2f x dx ln
, : i. f ii. f ,
.
42
f [ ]0,1 [ ]0,1 .
. ( )1
01f t dt
-
44
:f R R ( ) ( )204
1
x
f x dtf t
=+ , x R .
. f R . f ,
. ( ) ( )3 3 12f x f x x+ =. ( )f x x=
. ( )3
0
f x dx 45
[ ]: ,f a b R 0 a b< < ,
( )1z a if a= + ( )2z b if b= + 12
zw Rz
= .
. 1 2 1 2z iz z iz+ = . Rolle
( ) ( )f xg xx
= [ ],a b. f
. ( )
( )( )lim 1
x
x aa
f x a tdt
x a x a t+
= + , ( ) 1f x = ( ),a b
46
: (0, )f R+ ( ) 0f x 0x > .
( ) 11,1 , , 22
( )
2
1( )1 ( )
f xg x dt
f t=
: . ( )g x . ( )g x . 0x > ( ( ))f f x x= , :
i. '(1) 1f =
ii. (0,1)ox 2'( )
1o og x
x=
iii. (0,1) 3'( )2
f =
-
47
f : (0) 1f = ( ) ( )f x f x > [0, )x +
. [0, )x + ( ) xf x e>
. [0, )x + 0
( ) 2 3x
x xf t dt e+ +
(0,1) (1, ) +
. 0
( ) 2x
xf t dt e x+ = + (0,1)
48
f :[0, ) R+ , (0) 1f =
1
0
( ) ( ) (1) (1)x
f x f x f f edxe e
+ = .
. (0) 0f = . (0, )x + ( ) 1f x >
. (1) 2f = , 1
0
31 ( )2
f x dx<
iii. ( )2 2
( ) 1
1 1
ln ( )f xe dx f x dx >
49
: (0, )f R+ :
(1) 1f = 2 ( ( ) 1) ln 1x f x x = 0x >
. N ln( ) ( ) , 0xg x f x x
x= + > (0, )+ .
. N f .
-
. 1
11 xxx e + 1x . fC , 'x x
1,x x e= = .
50
( ) ln( 1) ln , 0xf x e x x= > . 0x > ( ) 0f x >. f .
. 2
1
lim ( )x
xx
f t dt+
++
= +
.
2
1
lim ( ) 0x
x
x
xf t dt+
=
51
f [1, )+ , ( ) 0f x > 1x
2
1
( ) ( ) , 1x
G x t f t dt x= 1
( ) ( ) , 1x
H x tf t dt x=
. ( )( )( )
G xF xH x
= (1, )+
. ( ) ( ) ( ), 1P x xH x G x x= . : i. 1x ( ) 0P x ii. P [1, )+
.
2
121
( ) lnlim
( )( 1)
x
x
H x tdt
G x x+
-
52
:f R R ( ) ( )1 2 1 2f x f x x + = + = , x R (1) 0f =
. ( ) ( )1 1 2 ,f x f x x x R+ + =
. ( ) ( )1
2
1
x
x
g x f t dt x+
=
. ( )3
1
I f x dx
=
53
( ) 22
x
f x u u du= .
. f. f ,
. ( ) ( )23 2
x ttg x e u u du dt
=
. ( )1 2, 1,3 ( ) ( ) ( )1 23
1 21
te f e e f t dt + =
. : ( )( )
3
3
33
xt f te dt x
f
( )2,x +
54
( ) 3xf x = 2( ) 9 5g x x x= + . . gC (1,4)A .
. 2( 5 6) ( ) 4f x x g x x + = .
. : ( 1) 5 ( ) 2lim( 1) ( ) 2
x
xx
f x f xAf x f x+
+ + =
+ + +
( 1) ( ) 2lim( 1) ( ) 2
x
xx
f x f xBf x f x
+ =
+ + +. f g .
. 2 1( )
( ) 1xI dx
f x
+=
+ R
-
55
f R ( )0 0f = ( ) ( )0
xF x f t dt=
. ( )F x R .. ( )F x .. ( ) 0F x .
. ( ) ( )3 1 1
0 01
x xf t dt x f t dt
+ ++ = + ( )0,1 .
56
. :f R .
f ( ) ( ) 0f x ,( ( ) 0f x ) x
. [ ], : 0,f g a R
0 0( ) ( )
a af t dt g t dt= , f [ ]0,a g [ ]0,a .
:
i. ( )( )
( ]0 , 0,
x
f t dtF x x a
x=
( )( )
0
x
g t dtG x
x=
.
ii. ( ) ( )0 0
x x
f t dt g t dt [ ]0,x a
iii. , [0, ]x y a 0 0
( ) ( )y x
x g t dt y f t dt
57
f [ ]0,e ( )0 0f = .
. ( ) ( ) ( )0
exf x dx ef e f e =
. ( )0,e ( ) ( ) ( )0
exf x dx e f e f =
. ( )1 ,e ( ) ( )( )10exf x dx ef e =
. [ ]2 0,e ( ) ( )2 20xxf x dx e f =
-
58
f R (1) 2f = ( ) 0f x > x R . :
. H ( ) ( )1
xg x f t dt= R (1, (1))A g .
. 1a > ( ) ( ) ( )1
0 11
aa f t dt f t dt
- http://www.mathematica.gr/forum/viewtopic.php?f=54&t=22769http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=129http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=129http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=2974
-
1
f R , 0x ( )0
0
2limh
f x hm R
h
= .
0( ) 2mf x =
2
f : R R
(2) 2f =0
( ) lim 33x
f xx= ( ) 0f x (0, 2)x
. (0) 0f = . (0) 9f = . f (0, (0))A f . ( ) 0f x = (0, 2) . (0, 2) ( ) 2f = . 1 2, (0, 2)x x 1 2( ) ( ) 1f x f x =
3
( )3 2
1 ,6 2
xx xf x x e x R
= + + +
.
. , , . . f
. 3 2
1,6 2
x x xe x x R + + +
. :g R R ( ) ( ) ( )3
0lim 1
6g x
x
g xe g x
=
. ( )
0lim 0x
g x
=
-
50
-
4
f R ( ) ( )( )2
f a ff x +
x . . 1 ( , )x a 1( ) 0f x =. ( ) ( ) 0f a f = = . ( , )a ( ) 0f = .
5
f R 3 ( ) 3 ( )f x f x x+ = x R . . f
. f 1f . f . f ,
. 0 < < , ( ) ( ) ( )f f f
>
. i. ( ) ( )g x f x x=
ii. 2 2( )f x x x x + y (1) 6, (1) 3f f = = . . ( )f x 0x > . . ( )f x (0, )+ . . f . . f (0, )+ f (1, (1))A f .
. , ,a R 1a = 103
a + + = , 2 2 2 1a + + > .
-
7
:[0, )f R+ , ( ) ( ( )) 2f x f f x x+ = [0, )x +
. ( ) 0f x [0, )x + (0) 0f = .
. ( ) ( ) 0
( ) ( ) 1
f xe f xf x f x
+ =
(0,1) .
. f 0 1x = f (1, (1))A f .
8
( ), : 0,f g R+ , f , ( )1 1f = ( ) 0f e = , ( ) ( )f x g xe x ce = + , 0x > c R. ( ) ( ) lnx f x x x x = +
Rolle [ ]1,e. , ( ) lnf = . c . f
9
1
*( ) ,xf x xe x R= . f . f . f
. 1xe
x
= , R
. ( )
20
ln 11limx
xf
x x +
-
10
f R (0) 0f =
0( )0 000
( 3 ) ( 2 )lim 10 f xh
f x h f x h x eh
+ = 0x R
. f . f .
11
2( ) ( 2) 2 3 2,xf x x e x x x R= + + . f f . f . . f . . f . ( ) 0f x = f . fC
12
( ) 1xf x e x= + . . f , .
. f ( 1) 1x xe x e a = a
. 2
1 , 02
x xe x x < + > .
. 2( ) ( )f x f x lnx= +
-
13
, :f g R R ( ) xf x e= 2( )g x x x= . . ( )f x (0,1)A ( )g x
. ( 1,0)a 2 1 0ae a+ + = . ( ) ( ) ( )h x f x g x= , :
i. 2( ) 1h x a a x Rii. ( ) 2012h x =
14
f ( ) 2 2xf x e x ex= + + . . f . . N ( ) 2f x =
15
:f R R *a R+ , :
( ) ( ) ( ) ( )2 lim2x a
a f x x f a a f a f ax a a
=
16
:f R R :g R R ln( ( )) ( )f x g x= x R . . f R. . :
i)( (1)) ( (2)) ( (3))lim 0( (4)) ( (2)) ( (3))
x x x
x x xx
f f ff f f+
+ +=
+
ii)( (1)) ( (2)) ( (3))lim( (4)) ( (2)) ( (3))
x x x
x x xx
f f ff f f
+ +=
+
. 1
( (1)) (1)lim 2 (1)1
x
x
f f gx
=
(1) 1f (1) ln 2g = .
. (1) 2f = lim ( )x
g x+
= + , 0 (1, )x + 0( ) 2012f x =
-
17
f 25 5( ) 2
5
x x
f x xln
= + .
. f 0 0x = .
. N 0 0x = .
18
:f R R 3
( ) 6lim 03x
f xx
=
5
(6 30)lim (5)5x
x fx
=
.
. ( ) 2f x x= + (3,5) . . f , ( ) 0f x = (3,5) .
19
( ) ( ) ( )3 33 ln 1 , 1,xf x x x x= + + + + ( ) 5 ln 2,xg x x x x R= + .. , , . . fC gC .
. , f
( )1,0 g ( )0,1
20
. 0r > 3 0lnr r+ =
. : (0, )f R+ 1( ) (1 )( 2)f x lnxx
=
i. ( )f x .ii. r () :
.2( 1)( ) 0rf x
r
+ 0x >
. ox r> ( ) ( ) 0o of x f x+ =
-
21
f : R R :
( ) 0f x , x R2
2
( 2) ( ) ( 4) lim 21 1x
x f x xx
+ =
.i. (2) 5f =
ii. 0 (1, 2)x , 0 0 0
1 1 20121 2 ( )x x f x+ =
. 2 2 2( ) ( ) 2f x f x x+ = , x R , : i. (1) 2f = ii. (1, (1))fiii. (1, 2) , ( 3) ( ) ( ) 1f f + =
22
. ( ) ( )xf x ln e x x= . , ;
. ( ) ( )xg x ln e x= ( ), ( )f x g x , 0ox x= > , 'y y
. ( )f x 1 2,r r :
( )( )1( )1 1 1) ( 2 1f rf r e r r = (1) 1 2 21
22
r r rer
=
(2)
23
( )f x , [0,1] : 2 22 ( ) (0) (1) 2f x f f + + [0,1]x
.i (0,1)c ( ) (1) (0)f c f f = ii. (0) 1f = , (1) 1f =
iii. ( )f x ( ) 2f x [0,1]xiv. 1 ( ) 1f x [0,1]x
A (0,1)t , : .i. 1 (0, )r t 1( ) ( ) 1tf r f t = +
ii. 2 ( ,1)r t 2( 1) ( ) ( ) 1t f r f t = iii. ( )f x [0,1]x
-
24
f [ ], . ( ) ( ) 0f f = = :. : ( ) ( ) 0f f < . f
. ( )0 ,x ( ) ( )0 0f x f x
25
( )2( ) ln 1f x x x= + + .. f ' . . fC .
. ( )lim
lnxf x
x+
. ( )( ) 2 3f x x f x= + .
26
f ( )11 1 1
4 2
x x
f x ln = +
.
. f 'x x. f .
. ( ) 1
142
f xx
ln +
0x < .
27
. ,h g ( ) ( )h x g x x 0x , 0 { }x R . :
i. 0 0
lim ( ) lim ( )x x x x
g x h x
= =
ii. 0 0
lim ( ) lim ( )x x x x
h x g x
= + = +
-
. :f R R :
( )2
22012 xxf x
x e =
+ x R .
i. f .ii. ( ) ( ) 2011g x f x x= Riii. f
iv. ( ) ( )lim 2x
f x f x+
+
28
:f R R ( ) 0f x > x R , lim ( ) 0x
f x+
= .
: . f . ( ) 0f x > x R .
29
:f R R ( ) ( ) ( )4xyf x y f x f y+ = ,x y R .. ( )0 1f =. ( )0 0f = , :
i. f R
ii. :g R R ( ) ( )22 xg x f x= Riii. f
30
f : (0, ) R+ ( ) ln ln , 0f x x a a x a= > . 0a > ( ) 0f x 0x > a e= : . f x ee x 0x >
. x ee x= , 0x > . , x x x x + + 0x >
http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=873 -
31
:f R R , :
( ) ( )( )2 f x f x x+ = x R
. f
. : ( ) ( ) , ,f x f y x y x y R
. : ( )1 1,2 2
x xf x x R +
. f 1f
32
( ): 0,f R+ , :( ) ( )2 2f xx e f x= + ( )0,x +
. ( )f x ( )f x ( )0,x +. f . f
. 1f
33
. lnxe x x> > ( )0,x +. ( ), 0,x t t= +
( ) xf x e= ( ) lng x x= ,A B .i. ( )AB ( )0,t + ( ) ( )AB d t=ii. ( ) 0d t =
( )0,1iii. d ( )0 0,1t
-
34
f ( ) ( ) 11 3xf x ln e x= + + .
. .
. ( )limx
f x+
( )limx
f x
.
. . . fC .
35
ln( ) , 0xf x x
x= >
. f . .
. R , 0xx e x= >
. ( )( ) xxx x = (0, )2
. (0, ) + fC
( , ( ))M f y y 2010
. x e 2 2ln ( 1) ln( 1) ( )
2x xf x f x+ + < + + +
38
:[1,f 2] R , f ( )x > 0 , x R .
( )( )
1 12 2
ifif
+=
+ 1. :
. ( )1 2f = ( )2 1f =. ( )f x x= ( )1,2. ( ) ( )f x f x x = ( )1,2
39
( ): 0,f R+
( ) ( ) ( )2 ln1 xx f x x = 0x >. (1)f . fC
. , R , g ( ), 0 1
( ), 0
f x xg x
x
+ <
-
40
, : (0, )f g R+ 2| ( ) ( ) |f x g x x xlnx+ 1x > . . + 1 2,e e ,
1 2,e e .
. 2
20
( 5 ) ( 3 ) 1(4
i)
l mh
f x h f x h x lnxsin h x
+ + =
1(1)2
f = , ( )f x
. ( )g x (1, 2)A , ;
41
[ ] [ ]: 1, 1, 4f e , [ ]1,4 , ( )1 2f = ( ) 1f e e= + . :.i. ( )1 2, 1,x x e 1 2x x ( ) ( )1 2 0f x f x = =
ii. ( )1,e ( ) 0f =iii. ( )1,ox e ( ) ( ) ( )( )23o o o of x f x f x x =
.i. : 2e x y e+ = + fC ( )1,oc eii. ( )1 2, 1,e ( )1 2 1,e ( ) ( )1 2 1f f =
42
f [ 2, 2] , ( 2, 2) , (0) 3f = ( ) ( ) ( )f x f x f x x = [ 2, 2]x
z 2z i = . :
. H f .
. 2 2( ) 2 ( ) 3 0f x f x x + = . H ( ) ( ) 1g x f x= ( 2, 2) . . H f
. 2( ) 1 4 , [ 2,2]f x x x= + . f z
z z , x x .
http://www.mathematica.gr/forum/viewtopic.php?f=55&t=9007 -
43
f [ ], , ( ), ( ) ( ) 0f f = = f ( ), .. ( ) 0f x > ( , )x . . i. 0 ( , )x f 0x .
ii. 1 2, ( , )x x 01 24 ( )( ) ( ) f xf x f x
.
. f ( ), , 1 2( , )x x
i. ( ) 0f ln2x x > 0
. 2
2xe x> x . ii.
2
1xx e+ =
. :f R R 2 ( )( ) f xf x e x+ = x R .
f fC 1x = .
45
:f R R : ( )( ) 1f xf x e x+ = + x R .
. ( 1)xe x + x R
. ( )2xf x x R lim ( )
xf x
=
. ( ) (1 )2xf x ln + 0x lim ( )
xf x
+= +
. ( )f x . .
. ( )f x 1( )f x
-
46
f R , : ( 1) 1, ( )f f > < ( 1) 1f + > + , R .
. , f 1o 3o . . , ( ) 1 ( )f x f x x = ( 1, 1) + . f R ( 1, 1) + ( ) 0f >
47
( ): 0,f R+ ( )1 0f = ( ) ( )2xf x f x x = ( )0,x + .
i. ( ) ( )2f x
h xx
= ( )0,x +
ii. f
iii. ( )g x R (1,0)
( ) ( )g x f x = x R . 21( )lim
lnxg x
x
48
f : (0, ) R+ (1) 1f = 2 ( )( ) f xf x
x = 0x > .
. 2( )f x x= 0x > . . M fC A M x x . A
(0,0)O , 2sec
o . 0t
M 3 , :
http://www.mathematica.gr/forum/viewtopic.php?f=55&t=9007 -
i. AM OM
ii. MOAiii. OB , B fC M x x
49
3 2( )P x ax bx cx d= + + + , , , , 0a b c d R a > 1 2 3, ,
. 2 3b ac> . . . 1 2,x x , : 1 2( ) ( ) 0P x P x + =
. 1 2,x x.
. 2( )( ) P xf x
x cx d=
+ + 2 25y x= +
1, 13x x= = 3 2( ) 2 12 13P x x x x= +
50
( ) : (0, )f x + , : 1
3 ( ) xx f x e = 0x > (1) , (1) 0f e f = =
. 1
( ) ( ) ( ) xg x xf x f x e= + (0, )+
. 1
( ) xf x xe= 0x >. ( )f x. :
i. ( )f x (2, (2))A f 1 ( ) ( )2
y e x e= +
ii. 1
2 ( 2)( )xxe x e + 0x >
- http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775E 30 (+ 10 ) 30 60 50